Harmonic conjugation in L1 on compact abelian groups

1992 ◽  
Vol 111 (1) ◽  
pp. 113-126
Author(s):  
Nakhl Asmar ◽  
Saleem Watson ◽  
Saleem Watson
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Abelian Groups ◽  
Duality Theorems ◽  
Character Group ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Torsion Free ◽  
Harmonic Conjugation

Let G denote a compact connected abelian group with character group and normalized Haar measure . As a consequence of the duality theorems (11, theorem 2518), is torsion-free and hence can be ordered. That is, there is a sub-semigroup P of such that

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

On the Upper Majorant Property for Locally Compact Abelian Groups

1978 ◽  
Vol 30 (5) ◽  
pp. 915-925
Author(s):  
M. Rains
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let G be a compact abelian group and form the spaces LP(G) with respect to the normalized Haar measure on G.

Sums of Complexes in Torsion Free Abelian Groups

1969 ◽  
Vol 12 (4) ◽  
pp. 479-480 ◽  
Author(s):  
H. Heilbronn ◽  
P. Scherk
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Preceding Paper ◽  
Torsion Free Abelian Group ◽  
Linearly Independent ◽  
Free Abelian Groups ◽  
Image Position ◽  
Torsion Free

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman
Keyword(s):  
Abelian Groups ◽  
Natural Order ◽  
Continuous Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Singular Measures ◽  
Compact Abelian Groups ◽  
Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

scholarly journals Abelian groups with small cotorsion images

1991 ◽  
Vol 50 (2) ◽  
pp. 243-247 ◽  
Author(s):  
Rüdiger Göbel
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Free Group ◽  
Compact Abelian Groups ◽  
Torsion Free

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

Cellular covers of ℵ1-free abelian groups

2015 ◽  
Vol 14 (10) ◽  
pp. 1550139 ◽  
Author(s):  
José L. Rodríguez ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Countable Rank ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Exact Sequences ◽  
Torsion Free ◽  
Cellular Cover

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

scholarly journals Fourier-Stieltjes Transforms which vanish at infinity off certain sets

1978 ◽  
Vol 19 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Measure Zero ◽  
Absolutely Continuous ◽  
Character Group ◽  
Convolution Algebra ◽  
Borel Measures ◽  
Discrete Measures ◽  
Stieltjes Transforms

In this paper G is a nondiscrete compact abelian group with character group Г and M(G) the usual convolution algebra of Borel measures on G. We designate the following subspaces of M(G) employing the customary notations: Ma(G) those measures which are absolutely continuous with respect to Haar measure; MS(G) the space of measures concentrated on sets of Haar measure zero and Md(G) the discrete measures.

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